The usual equation for a parabola looks something like y = -x^2. If that was a projectile following a parabolic path, you could find the height (y) of the bullet if you knew the position (x). If you wanted to find out where the bullet was 3 seconds after firing, you would be out of luck.

The easiest way to think of parametric equations is to treat 't' like time. Taking the equation above, let's say that the bullet is traveling at 3 units/second. So you could express x like so:
x = 3*t
Now you how far the bullet has went when given the time (t). Putting the two equations together you have this:
x = 3*t
y = -x^2
Which is equivalent to the parametric form if you solve y in terms of t:
x = 3*t
y = -9*t^2

Parametric equations are just another tool to simplify the math. The ray/plane collision code uses it because it's very easy to solve for t when the ray is expressed as a parametric form. Once you solve for the t, it's very easy to find the x, y, and z of the collision.

D is the distance of the plane from the origin (0,0,0) in direction of the plane's normal:
D = (any point on the plane) • (the plane's normal)
Where '•' is the dot product.

To answer your second question:
Lines are infinite in both directions so t can be anywhere in the range [-infinity, infinity].
Rays are infinite in only one direction so t must be in the range [0, infinity].
Line segments are generally defined so that they cover the range [0, 1] by defining them as Segment(t) = p0 + (p1 - p0)*t

So after solving for t, just check that it falls in the correct range. It sounds like you are really after line segment collisions. Though as you can see, the math is pretty much identical.